Hamiltonian phase, complete-domain flow and reference measure

Assume countable choice. Let $M$ be a Hausdorff second-countable smooth $n$-manifold and let $\eta$ be a positive smooth density, with Borel coefficients as in Pointwise Borel nonnegative densities. Let ${\mu }_{\eta }$ be the intrinsic chart-gluing Borel measure of Intrinsic density measure and its chart restriction. If $\varphi :M\to M$ is a $C^1$ diffeomorphism satisfying ${\varphi }^{\ast }\eta =\eta$, then ${\mu }_{\eta }({\varphi }^{-1}(E))={\mu }_{\eta }(E)$ for every Borel set $E\subseteq M$. The total measure may be infinite.

The coordinate comparison used in step 2.1 is the commuting square below. The proof uses its coordinate-map Jacobian through [F2]; the diagram is only an aid to locating that map.

S_{ij}\subseteq U_i \arrow[r,"\phi"] \arrow[d,"x_i"'] &
\phi(S_{ij})\subseteq V_j \arrow[d,"y_j"] \\
x_i(S_{ij}) \arrow[r,"y_j\circ\phi\circ x_i^{-1}"'] &
y_j(\phi(S_{ij}))

Adopt ${\mathrm{A}\mathrm{C}}_{\omega }$ for the density-measure construction (The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$)). A Hamiltonian phase datum consists of a Hausdorff second-countable smooth manifold $M$ of finite even dimension $2n$ without boundary, a symplectic form $\omega$ on $M$, and an autonomous smooth function $H:M\to \mathbb{R}$. Here symplectic means that $\omega$ is a smooth closed two-form and is nondegenerate at every point (Symplectic form and symplectic manifold). The Hamiltonian vector field $X_H$ is the unique smooth field satisfying ${\iota }_{X_H}\omega =dH$ (Hamiltonian vector field and Hamiltonian function, Hamiltonian vector fields exist uniquely for smooth functions). For each $z\in M$, let $I_z\subseteq \mathbb{R}$ be the maximal open interval containing $0$ on which the unique maximal $C^{\mathrm{\infty }}$ integral curve ${\gamma }_z:I_z\to M$ satisfying ${\dot{\gamma }}_z(t)=X_H({\gamma }_z(t))$ and ${\gamma }_z(0)=z$ is defined; its existence and uniqueness follow from the smooth ODE theorem (The fundamental theorem for autonomous smooth ODEs). Set $D=\{(t,z):z\in M,t\in I_z\}$ and define the maximal local flow $\mathrm{\Phi }:D\to M$ by $\mathrm{\Phi }(t,z)={\gamma }_z(t)$. Write ${\mathrm{\Phi }}_t(z)=\mathrm{\Phi }(t,z)$ when $(t,z)\in D$. Uniqueness gives ${\mathrm{\Phi }}_0(z)=z$ and ${\mathrm{\Phi }}_t({\mathrm{\Phi }}_s(z))={\mathrm{\Phi }}_{t+s}(z)$ whenever the terms are defined. The domain $D$ is open, and in local charts $\mathrm{\Phi }$ is smooth in the initial state and $C^1$ in time, as supplied by the stated ODE theorem. Completeness of the flow on all of $M$ is not part of this datum.

A Borel set $R\subseteq M$ is a complete invariant domain for this datum if $\{t\}\times R\subseteq D$ and ${\mathrm{\Phi }}_t(R)=R$ for every $t\in \mathbb{R}$. Thus each restricted map ${\mathrm{\Phi }}_t{\mathrm{\mid }}_R:R\to R$ is defined everywhere on $R$. The definition makes no assertion that $M$ itself is complete or that such an $R$ has finite measure.

The symplectic orientation has volume form $\mathrm{\Omega }={\omega }^n\mathrm{/}n!$ (Symplectic manifolds have a canonical orientation and volume form). For a fixed positive reference action $a_0$, define the positive smooth density ${\eta }_0=\mathrm{\mid }\mathrm{\Omega }\mathrm{\mid }\mathrm{/}{a}_0^n$ and let $m_0$ be its intrinsic Borel measure (Density bundle and smooth density fields, Pointwise Borel nonnegative densities, Intrinsic density measure and its chart restriction, Positive smooth densities give Radon volume). The measure $m_0$ may have infinite total mass. Use SI units for a physical interpretation. In a standard mechanical cotangent chart, $q^i$ has unit m, $p_i$ has unit kg m/s, $H$ has unit J, $\omega$ and $a_0$ have unit J s, and ${\eta }_0$ is dimensionless. The scale $a_0$ is a normalization convention and does not assert quantization.

An ensemble on $R$ is a Borel probability measure $\mu$ on $R$, distinct from an individual trajectory. If $\mu$ has density $\rho$ with respect to $m_0{\mathrm{\mid }}_R$, then $\rho \ge 0$ is Borel and ${\int }_R\rho d{m}_0=1$. For $\beta >0$, the canonical probability, when it exists, is the Borel measure ${\mu }_{\beta }(B)=Z_{\beta }^{-1}{\int }_B{e}^{-\beta H}d{m}_0,Z_{\beta }={\int }_R{e}^{-\beta H}d{m}_0,$ for Borel $B\subseteq R$, provided $0<Z_{\beta }<\mathrm{\infty }$. For a temperature-parameterized physical interpretation, take $T>0$ as an absolute temperature in K and $k_B>0$ as a supplied energy-temperature conversion constant in J/K, and set $\beta =(k_{B}T{)}^{-1}$ in J${}^{-1}$.

The flow's actual domain, its complete invariant subsets, the reference measure, and any ensemble probability are separate data. Smoothness alone does not imply completeness, finite phase volume, normalizability of a canonical weight, recurrence, or ergodicity.

Let $(\mathrm{\Xi },\mathcal{F},\mathbb{P})$ be a probability space and let $V:\mathbb{R}\times \mathrm{\Xi }\to {\mathbb{R}}^d$ be jointly measurable for the completed product of Lebesgue measure and $\mathbb{P}$. Assume $\mathbb{E}[V(t)]=0$ for every $t$, $\mathbb{E}\mathrm{\mid }V(0){\mathrm{\mid }}^2<\mathrm{\infty }$, and second-order stationarity in the explicit sense that each covariance below is finite and $C_{ij}(s):=\mathbb{E}[V_i(u)V_j(u+s)]$ is independent of $u\in \mathbb{R}$ for every $s\in \mathbb{R}$ and every $1\le i,j\le d$. For $T>0$, the Lebesgue integral $X(T)={\int }_0^{T}V(s)ds$ exists almost surely and in $L^2$. Define $\mathrm{Cov}(X(T){)}_{ij}=\mathbb{E}[(X_i(T)-\mathbb{E}{X}_i(T))(X_j(T)-\mathbb{E}{X}_j(T))];$ the centered hypothesis makes these means zero. Then $\mathrm{Cov}(X_i(T),X_j(T))={\int }_0^T(T-s)(C_{ij}(s)+C_{ji}(s))ds\mathrm{.}$ If each $C_{ij}$ is absolutely integrable on $[0,\mathrm{\infty })$, then $\frac{\mathrm{Cov}(X(T))}{2T}⟶D,D_{ij}=\frac{1}{2}{\int }_0^{\mathrm{\infty }}(C_{ij}(s)+C_{ji}(s))ds,$ and $D$ is positive semidefinite. This is a variance-growth conclusion; it does not assert a central limit theorem or identify a transport coefficient.

The MIT notes derive a self-diffusion formula from a physical long-time mean-square-displacement assumption. The theorem here proves only the stated covariance limit under second-order stationarity and absolute integrability; it supplies neither that physical identification nor a Gaussian limit.

Let $(X,\mathcal{A},\mu ,T)$ be a probability-preserving system: $\mu (X)=1$, $T:X\to X$ is measurable, and $\mu (T^{-1}E)=\mu (E)$ for every $E\in \mathcal{A}$ (Measure-preserving transformations and systems). The ergodicity convention is that every strictly invariant set $E\in \mathcal{A}$, meaning $T^{-1}E=E$, has measure zero or one, as in Ergodicity relative to an invariant measure.

For $E,F\in \mathcal{A}$ and integer $n\ge 0$, put $d_n(E,F)=\mu (E\cap T^{-n}F)-\mu (E)\mu (F)\mathrm{.}$ The map is strongly mixing if $d_n(E,F)\to 0$ for every measurable pair $(E,F)$. It is weakly mixing if $\frac{1}{N}{\sum }_{n=0}^{N-1}\mathrm{\mid }{d}_n(E,F)\mathrm{\mid }⟶0$ for every such pair as $N\to \mathrm{\infty }$. Thus the absolute value is inside the Cesàro average, exactly as in Strong and weak mixing on a probability space. These are properties of the selected discrete map $T$ and its selected invariant probability $\mu$; no flow-level property is implicit.

For a selected finite-dimensional nonrelativistic classical model, adopt a Hamiltonian phase datum Hamiltonian phase, complete-domain flow and reference measure and interpret each point of its manifold $M$ as a complete instantaneous model state. The Hamiltonian $H:M\to \mathbb{R}$ is the model's supplied primitive energy function; no derivation from a Lagrangian is assumed here. Choose a complete invariant domain $R$ and a prepared Borel probability measure ${\mu }_0$ on $R$. For each initial state $z_0\in R$, the unforced state curve is the smooth map $z_{z_0}:\mathbb{R}\to M$ defined by $z_{z_0}(t)={\mathrm{\Phi }}_t(z_0)$ using the flow from the phase datum; complete invariance ensures $z_{z_0}(t)\in R$ for every $t$. The preparation is a probability law over possible phase points; it is not an individual trajectory or a claim that an ensemble was experimentally sampled.

Use SI units for the physical interpretation. In a canonical cotangent chart, $q^i$ are generalized configuration coordinates and $p_i$ are the conjugate covector components, so ${\sum }_i{p}_{i}d{q}^i$ has units J s. If $q^i$ has unit $u_i$, then $p_i$ has unit J s/$u_i$. Thus Cartesian length coordinates have units m and their conjugates kg m/s, whereas dimensionless angular coordinates have conjugates in J s. Identifying a conjugate with mechanical momentum requires an additional model convention. The Hamiltonian has units J and time has units s. The reference action $a_0$ in the mathematical definition has units J s and is a normalization convention. If a canonical-temperature model is additionally selected, take $T>0$ as absolute temperature in K and adopt $k_B>0$ as a primitive energy-temperature conversion constant of this formulation, with units J/K. Set $\beta =(k_{B}T{)}^{-1}$ in J${}^{-1}$ and use the associated weight only when its partition integral is finite and positive. This postulate does not assert that the canonical law is normalizable for every Hamiltonian or that these adopted parameters describe nature.

This is an adopted model formulation and preparation rule. It is not derived from mathematical axioms, and citing statistical-mechanics treatments does not establish that a selected model describes nature. No ergodicity, equilibration, measured outcome, or stochastic bath is assumed.

Let $(M,\omega ,H,\mathrm{\Phi },m_0)$ be a nonempty compact symplectic phase datum without boundary, with $H\in C^{\mathrm{\infty }}(M,\mathbb{R})$. Let $A,B\in C^{\mathrm{\infty }}(M,\mathbb{R})$ be observables, let $\beta >0$, and assume $Z_{\beta }={\int }_M{e}^{-\beta H}d{m}_0$ is finite and positive. Put $d{\mu }_{\beta }=Z_{\beta }^{-1}{e}^{-\beta H}d{m}_0$. For $T>0$ and every $t\in [0,T]$, choose $h\in C^{\mathrm{\infty }}([0,T],\mathbb{R})$ and perturb the Hamiltonian to $H_{ϵ}(t,z)=H(z)-ϵh(t)B(z)$, where $ϵ$ is dimensionless. In SI, time has unit s. If $H$ is in J, $A$ has units $[A]$, and $B$ has units $[B]$, then $h$ has units J$\mathrm{/}[B]$, $\beta$ has units J${}^{-1}$, and the response derivative has units $[A]$.

Start every perturbed trajectory at the same unperturbed initial law ${\mu }_{\beta }$ and define $⟨A(t){⟩}_{ϵ}={\int }_{M}A(z_{ϵ}(t;z))d{\mu }_{\beta }(z)$. Then the derivative at $ϵ=0$ exists and is ${\frac{d}{dϵ}⟨A(t){⟩}_{ϵ}\mid }_{ϵ=0}={\int }_0^{t}h(s)R_{AB}(t-s)ds,R_{AB}(u)=\beta {\int }_M(X_{H}B)(z)A({\mathrm{\Phi }}_{u}z)d{\mu }_{\beta }(z)\mathrm{.}$ For constant $h(t)=h_0$, define the equilibrium correlation $C_{BA}(u)={\int }_M(B-⟨B{⟩}_{\beta })(z)(A-⟨A{⟩}_{\beta })({\mathrm{\Phi }}_{u}z)d{\mu }_{\beta }(z)\mathrm{.}$ Then the finite-time response is $\beta h_0[C_{BA}(0)-C_{BA}(t)]$. If $C_{BA}(t)\to 0$, its long-time limit is $\beta h_0{\mathrm{Cov}}_{{\mu }_{\beta }}(B,A)$. No decay of this correlation is assumed in the finite-time formula.

The conclusion is a mathematical model prediction conditional on its compact phase, smooth-control, and canonical-preparation hypotheses. The cited statistical-mechanics treatments motivate response/correlation formulas; the proof above supplies this precise finite-time sign convention and does not extend it to noncompact phases or unproved decay assumptions.

Let $(M,\omega ,H,\mathrm{\Phi },m_0)$ be a Hamiltonian phase datum, let $R\subseteq M$ be a complete invariant Borel domain, and fix $\mathrm{\Delta }>0$. Put $T={\mathrm{\Phi }}_{\mathrm{\Delta }}{\mathrm{\mid }}_R$. Let $\mu$ be any finite Borel measure on $R$ invariant under $T$. Then for every Borel $E\subseteq R$, $\mu$-almost every $z\in E$ satisfies $T^{k}z\in E$ for infinitely many integers $k\ge 1$. The result is almost-everywhere recurrence for this fixed sampled map; it is not a pointwise statement about every initial state. The hypothesis also allows finite invariant measures singular with respect to ambient Liouville volume.

The theorem does not assume or prove finite total Liouville volume for $M$. An infinite-volume phase space can still have a finite invariant probability or a finite invariant restriction, but that is a separate hypothesis.

Let $(M,\omega ,H,\mathrm{\Phi },m_0)$ be a Hamiltonian phase datum and let $R$ be a complete invariant Borel domain. If $w:\mathbb{R}\to [0,\mathrm{\infty }]$ is Borel and $0<Z_w:={\int }_{R}w(H)d{m}_0<\mathrm{\infty },$ then ${\mu }_w(B)=Z_w^{-1}{\int }_{B}w(H)d{m}_0$ is an invariant Borel probability on $R$. In particular, for $\beta >0$, the canonical density $e^{-\beta H}\mathrm{/}{Z}_{\beta }$ defines an invariant probability whenever $Z_{\beta }={\int }_R{e}^{-\beta H}d{m}_0$ is finite and positive.

There is also the following regular-shell conclusion. Assume full AC and $n\ge 1$. Let $O\subseteq {\mathbb{R}}^{2n}$ be open with canonical $\omega ={\sum }_{i}d{q}^i\wedge d{p}_i$, let $H\in C^2(O,\mathbb{R})$, and let $J\subseteq \mathbb{R}$ be open. Suppose $\mathrm{\nabla }H\mathrm{\ne }0$ on $H^{-1}(J)$ and $H^{-1}(K)$ is compact in $O$ for every compact $K\subset J$. For each energy $E\in J$ with nonempty shell ${\mathrm{\Sigma }}_E=H^{-1}(\{E\})$, the weighted area measure ${\mu }_E$ of Regular energy shell and weighted area measure satisfies $0<{\mu }_E(1)<\mathrm{\infty }$, and ${\widehat{\mu }}_E={\mu }_E\mathrm{/}{\mu }_E(1)$ is invariant under every Hamiltonian flow diffeomorphism preserving a full compact band containing ${\mathrm{\Sigma }}_E$. Critical levels and noncompact bands are outside this clause.

The two clauses have different normalizability requirements: the energy weight must have a finite positive partition integral, while the shell clause requires the exact compact regular-band hypotheses above. Neither follows from Hamiltonian volume preservation alone.

Let $(M,\omega ,H,\mathrm{\Phi },m_0)$ be a Hamiltonian phase datum and let $R$ be a complete invariant Borel domain carrying a Borel probability $\mu$ such that $\mu ({\mathrm{\Phi }}_s^{-1}B)=\mu (B)$ for every $s\in \mathbb{R}$ and every Borel $B\subseteq R$. Fix a sampling interval $\mathrm{\Delta }>0$ and define the time map $T={\mathrm{\Phi }}_{\mathrm{\Delta }}{\mathrm{\mid }}_R$. Assume this particular map is ergodic for $\mu$, meaning that every Borel $E\subseteq R$ with $T^{-1}E=E$ has $\mu (E)\in \{0,1\}$. For every real $f\in L^1(\mu )$, the sampled averages $\frac{1}{N}{\sum }_{j=0}^{N-1}f(T^{j}z)$ converge $\mu$-almost surely and in $L^1(\mu )$ to ${\int }_{R}fd\mu$. If $f$ is additionally bounded and Borel, then the continuous-time averages $\frac{1}{\tau }{\int }_0^{\tau }f({\mathrm{\Phi }}_{s}z)ds$ converge $\mu$-almost surely and in $L^1(\mu )$ to the same mean as $\tau \to \mathrm{\infty }$. No ergodicity of every time map or generic Hamiltonian flow is asserted.

The hypothesis is about the chosen sampled map ${\mathrm{\Phi }}_{\mathrm{\Delta }}$. A continuous flow may be ergodic while a resonant sampled time map is not, so the theorem does not replace the explicit ergodicity assumption.

For one fixed probability-preserving map $T$, strong mixing implies weak mixing, and weak mixing implies ergodicity, under the exact definitions in Ergodicity and mixing for a selected probability map. No converse is asserted.

The implications concern only the selected probability-preserving map. Hamiltonian volume preservation by itself does not supply either mixing hypothesis.

Let $(M,\omega ,H,\mathrm{\Phi },m_0)$ be a Hamiltonian phase datum and let $R\subseteq M$ be an open complete invariant domain. For any finite-valued Borel density ${\rho }_0\ge 0$ with ${\rho }_0\in L^1(m_0{\mathrm{\mid }}_R)$ and ${\int }_R{\rho }_{0}d{m}_0=1$, define ${\rho }_t={\rho }_0\circ {\mathrm{\Phi }}_{-t}$. Then ${\rho }_t$ is a probability density for every $t$, and for every bounded Borel observable $A$, ${\int }_{R}A{\rho }_{t}d{m}_0={\int }_R(A\circ {\mathrm{\Phi }}_t){\rho }_{0}d{m}_0\mathrm{.}$ If ${\rho }_0$ is smooth, it satisfies the classical Liouville equation ${\mathrm{\partial }}_t{\rho }_t+X_H{\rho }_t=0$. For merely $L^1$ data, it satisfies the distributional equation on $R$, meaning ${\int }_{\mathbb{R}}{\int }_R{\rho }_t({\mathrm{\partial }}_t\phi +X_H\phi )d{m}_{0}dt=0$ for every smooth compactly supported test function $\phi \in C_c^{\mathrm{\infty }}(\mathbb{R}\times R)$.